Statistics for Business and Economics 7th Edition Chapter 15 Analysis of Variance Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-1 Chapter Goals After completing this chapter, you should be able to: Recognize situations in which to use analysis of variance Understand different analysis of variance designs Perform a one-way and two-way analysis of variance and interpret the results Conduct and interpret a Kruskal-Wallis test Analyze two-factor analysis of variance tests with more than one observation per cell Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-2 15.2 One-Way Analysis of Variance Evaluate the difference among the means of three or more groups Examples: Average production for 1st, 2nd, and 3rd shifts Expected mileage for five brands of tires Assumptions Populations are normally distributed Populations have equal variances Samples are randomly and independently drawn Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-3 Hypotheses of One-Way ANOVA H0 : μ1 = μ2 = μ3 = = μK All population means are equal i.e., no variation in means between groups H1 : μi ≠ μ j for at least one i, j pair At least one population mean is different i.e., there is variation between groups Does not mean that all population means are different (some pairs may be the same) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-4 One-Way ANOVA H0 : μ1 = μ2 = μ3 = = μK H1 : Not all μi are the same All Means are the same: The Null Hypothesis is True (No variation between groups) μ1 = μ2 = μ3 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-5 One-Way ANOVA (continued) H0 : μ1 = μ2 = μ3 = = μK H1 : Not all μi are the same At least one mean is different: The Null Hypothesis is NOT true (Variation is present between groups) or μ1 = μ2 ≠ μ3 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall μ1 ≠ μ2 ≠ μ3 Ch 15-6 Variability The variability of the data is key factor to test the equality of means In each case below, the means may look different, but a large variation within groups in B makes the evidence that the means are different weak A B A B Group C Small variation within groups Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall A B Group C Large variation within groups Ch 15-7 Partitioning the Variation Total variation can be split into two parts: SST = SSW + SSG SST = Total Sum of Squares Total Variation = the aggregate dispersion of the individual data values across the various groups SSW = Sum of Squares Within Groups Within-Group Variation = dispersion that exists among the data values within a particular group SSG = Sum of Squares Between Groups Between-Group Variation = dispersion between the group sample means Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-8 Partition of Total Variation Total Sum of Squares (SST) = Variation due to random sampling (SSW) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall + Variation due to differences between groups (SSG) Ch 15-9 Total Sum of Squares SST = SSW + SSG K ni SST = ∑∑ (x ij − x) i=1 j=1 Where: SST = Total sum of squares K = number of groups (levels or treatments) ni = number of observations in group i xij = jth observation from group i x = overall sample mean Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-10 Two-Way Notation Let xji denote the observation in the jth group and ith block Suppose that there are K groups and H blocks, for a total of n = KH observations Let the overall mean be x Denote the group sample means by x j• (j = 1,2, ,K) Denote the block sample means by x •i (i = 1,2, ,H) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-43 Partition of Total Variation SST = SSG + SSB + SSE Total Sum of Squares (SST) = Variation due to differences between groups (SSG) + Variation due to differences between blocks (SSB) + The error terms are assumed to be independent, normally distributed, and have the same variance Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Variation due to random sampling (unexplained error) (SSE) Ch 15-44 Two-Way Sums of Squares The sums of squares are Total : Degrees of Freedom: K H SST = ∑∑ (x ji − x)2 n–1 j=1 i =1 Between - Groups : K SSG = H∑ (x j• − x)2 K–1 j=1 Between - Blocks : H SSB = K ∑ (x •i − x)2 H–1 i =1 Error : K H SSE = ∑∑ (x ji − x j• − x •i + x)2 (K – 1)(K – 1) j =1 i =1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-45 Two-Way Mean Squares The mean squares are SST MST = n −1 SST MSG = K −1 SST MSB = H −1 SSE MSE = (K − 1)(H − 1) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-46 Two-Way ANOVA: The F Test Statistic H0: The K population group means are all the same H0: The H population block means are the same Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall F Test for Groups MSG F= MSE Reject H0 if F > FK-1,(K-1)(H-1),α F Test for Blocks MSB F= MSE Reject H0 if F > FH-1,(K-1)(H-1),α Ch 15-47 General Two-Way Table Format Source of Variation Between groups Sum of Squares Degrees of Freedom SSG K–1 Between blocks SSB H–1 Error SSE (K – 1)(H – 1) Total SST n-1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Mean Squares MSG = MSB = MSE = SSG K −1 SSB H −1 F Ratio MSG MSE MSB MSE SSE (K − 1)(H − 1) Ch 15-48 More than One Observation per Cell 15.5 A two-way design with more than one observation per cell allows one further source of variation The interaction between groups and blocks can also be identified Let K = number of groups H = number of blocks L = number of observations per cell n = KHL = total number of observations Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-49 More than One Observation per Cell SST = SSG + SSB + SSI + SSE SSG Between-group variation SST Total Variation SSB Between-block variation SSI n–1 (continued) Degrees of Freedom: K–1 H–1 Variation due to interaction between groups and blocks (K – 1)(H – 1) SSE KH(L – 1) Random variation (Error) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-50 Sums of Squares with Interaction Degrees of Freedom: Total : SST = ∑∑∑ (x jil − x)2 j Between - groups : i l K SSG = HL∑ (x j•• − x)2 j=1 Between - blocks : n-1 K–1 H SSB = KL ∑ (x •i• − x)2 H–1 i=1 Interaction : K H SSI = L ∑∑ (x ji• − x j•• − x •i• + x)2 j=1 i=1 Error : SSE = ∑∑∑ (x jil − x ji• )2 i Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall j l (K – 1)(H – 1) KH(L – 1) Ch 15-51 Two-Way Mean Squares with Interaction The mean squares are MST = SST n −1 MSG = SST K −1 MSB = SST H −1 MSI = SSI (K - 1)(H − 1) SSE MSE = KH(L − 1) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-52 Two-Way ANOVA: The F Test Statistic H0: The K population group means are all the same H0: The H population block means are the same H0: the interaction of groups and blocks is equal to zero Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall F Test for group effect MSG F= MSE Reject H0 if F > FK-1,KH(L-1),α F Test for block effect MSB F= MSE Reject H0 if F > FH-1,KH(L-1),α F Test for interaction effect MSI F= MSE Reject H0 if F > F(K-1)(H-1),KH(L-1),α Ch 15-53 Two-Way ANOVA Summary Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Statistic Between groups SSG K–1 MSG = SSG / (K – 1) MSG MSE MSB MSE MSI MSE Between blocks SSB H–1 MSB = SSB / (H – 1) Interaction SSI (K – 1)(H – 1) MSI = SSI / (K – 1)(H – 1) Error SSE KH(L – 1) MSE = SSE / KH(L – 1) Total SST n–1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-54 Features of Two-Way ANOVA F Test Degrees of freedom always add up n-1 = KHL-1 = (K-1) + (H-1) + (K-1)(H-1) + KH(L-1) Total = groups + blocks + interaction + error The denominator of the F Test is always the same but the numerator is different The sums of squares always add up SST = SSG + SSB + SSI + SSE Total = groups + blocks + interaction + error Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-55 Examples: Interaction vs No Interaction Interaction is present: No interaction: Block Level Block Level A B Groups C Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Mean Response Mean Response Block Level Block Level Block Level Block Level A B Groups C Ch 15-56 Chapter Summary Described one-way analysis of variance The logic of Analysis of Variance Analysis of Variance assumptions F test for difference in K means Applied the Kruskal-Wallis test when the populations are not known to be normal Described two-way analysis of variance Examined effects of multiple factors Examined interaction between factors Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-57 ... Hall Ch 15- 2 15. 2 One-Way Analysis of Variance Evaluate the difference among the means of three or more groups Examples: Average production for 1st, 2nd, and 3rd shifts Expected mileage for five.. .Chapter Goals After completing this chapter, you should be able to: Recognize situations in which to use analysis of variance Understand different analysis of variance designs Perform... as Prentice Hall Ch 15- 30 Multiple Supgroups: Example x1 = 249.2 n1 = x2 = 226.0 n2 = x3 = 205.8 n3 = Sp 93.3 MSD(k) = q = 3.77 = 9.387 n 15 (where q = 3.77 is from Table 13 for α = 05 and 12 df)